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202405101842 Exercise 16.67

The cross product of a vector with itself is

A. the vector \mathbf{i}^2+\mathbf{j}^2+\mathbf{k}^2.
B. the vector \mathbf{i}+\mathbf{j}+\mathbf{k}.
C. the zero vector.
D. the scalar quantity 1.
E. the scalar quantity 0.

Extracted from Stan Gibilisco. (2006). Technical Math Demystified.

Erratum.

Option A should be a scalar.

Roughwork.

Most commonly, it is the three-dimensional Euclidean space \mathbf{E}^3 (or \mathbb{E}^3) that models physical space \mathbf{R}^3 (or \mathbb{R}^3).

Wikipedia on Three-dimensional space

The Euclidean metric is given by

\mathbf{G}=(g_{\alpha\beta})\equiv \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{bmatrix}

and so line element \mathrm{d}s

\begin{aligned} \mathrm{d}s^2 & = g_{\alpha\beta}\,\mathrm{d}x^\alpha\,\mathrm{d}x^\beta \\ & = (\mathrm{d}x^1)^2 + (\mathrm{d}x^2)^2 + (\mathrm{d}x^3)^2 \\ & = \mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2 \\ \end{aligned}

or distance

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

is equivalent to applying twice the Pythagorean equation

a^2+b^2=c^2

to two points (x_1,y_1,z_1) and (x_2,y_2,z_2) with respect to the standard basis (\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z):

\begin{cases} \hat{\mathbf{i}}=\mathbf{e}_x=(1,0,0) \\ \hat{\mathbf{j}}=\mathbf{e}_y=(0,1,0) \\ \hat{\mathbf{k}}=\mathbf{e}_z=(0,0,1) \\ \end{cases}

of a Cartesian coordinate system. To carry a point from one position to another, a vector \mathbf{v} has both magnitude v=|\mathbf{v}| and direction \hat{\mathbf{v}}=\frac{\mathbf{v}}{|\mathbf{v}|}. Scalar multiplication c\mathbf{v} will scale its magnitude by some factor c (the scalar) without change in its direction. I.e.,

\begin{aligned} \mathbf{u} & \stackrel{\mathrm{def}}{=}c\mathbf{v} \\ u & = |\mathbf{u}| = |c\mathbf{v}| = c|\mathbf{v}|\\ & = cv \\ \hat{\mathbf{u}} & = \frac{\mathbf{u}}{|\mathbf{u}|} = \frac{c\mathbf{v}}{c|\mathbf{v}|} = \frac{\mathbf{v}}{|\mathbf{v}|} \\ & = \hat{\mathbf{v}} \\ \end{aligned}

Vector multiplication is for whose vectors be the products scalar or vector. For instance, we have

Dot product (aka scalar product) of two vectors \mathbf{u}=(u_x,u_y,u_z) and \mathbf{v}=(v_x,v_y,v_z)

\begin{aligned} \mathbf{u}\cdot\mathbf{v} & = \sum_{x,y,z}u_iv_i \\ & = u_xv_x+u_yv_y+u_zv_z \\ & = |\mathbf{u}| |\mathbf{v}|\cos\theta \\ \end{aligned}

as a scalar quantity; but their cross product (aka vector product)

\begin{aligned} \mathbf{u}\times\mathbf{v} & = \begin{vmatrix} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \\ \end{vmatrix} \\ & = (u_yv_z-u_zv_y, u_zv_x-u_xv_z, u_xv_y-u_yv_x) \\ & = |\mathbf{u}||\mathbf{v}|\sin\theta\,\hat{\mathbf{n}} \\ \end{aligned}

a vector quantity. Added on are tensor product \mathbf{u}\otimes\mathbf{v}, wedge product \mathbf{u}\wedge\mathbf{v}, and more.

This problem is not to be attempted.